Transcript of MinshengLi dissertation final02
MinshengLi_dissertation_final02.docAbstract LI, MINSHENG. Nonlinear
Behavioral Modeling of Quadrature Modulators and Analysis of
Impacts on Wireless Communication Systems. (Under the Direction of
Professor Kevin G. Gard.)
Direct-conversion quadrature modulators have been widely used in
modern
communication systems because of their capability to conserve
bandwidth with low cost,
low complexity, and small form factor. In this dissertation, three
different quadrature
modulator applications: 1) generic quadrature modulation system; 2)
zero-IF OFDM
system; and 3) low-IF OFDM system are reviewed. The different
usages and
implementation costs of the modulators are compared. Impacts of
carrier leakage,
gain/phase imbalances, and nonlinear distortion on system
performance degradation in
terms of spectrum regrowth and waveform quality are analyzed for
three applications of
the quadrature modulators. The adverse impacts of these physical
imperfections demand
system-level techniques to accurately characterize them for system
design and verification.
A new nonlinear behavioral model is developed for characterization
of both
correlated/uncorrelated nonlinear distortion and the linear static
errors including DC offset
and gain/phase imbalances for direct-conversion quadrature
modulators. This enables fast
and accurate prediction of spectrum regrowth and waveform quality
degradation due to
these physical imperfections. A low-pass equivalent model structure
is developed based on
the assumptions that I and Q channels are isolated and the dominant
nonlinearities of
quadrature modulators are the baseband transconductors. The
responses of I and Q inputs
are modeled by two independent complex power series. These capture
both nonlinear
distortion and linear static errors. Amplitude-to-amplitude (AM-AM)
and
amplitude-to-phase (AM-PM) measurements and 4-point vector network
analyzer (VNA)
measurements are used to extract the model parameters for
characterization of the
nonlinear distortion and the linear static errors, respectively. An
existing orthogonalization
technique for power amplifiers is implemented in the quadrature
modulator model to
decompose the correlated and uncorrelated nonlinear distortion. The
modeling technique is
applied to both passive and active RF quadrature modulators and the
models are verified by
adjacent channel power ratio (ACPR) and error vector magnitude
(EVM) measurements of
systems excited by digitally modulated signals.
One fundamental quadrature modulator model assumption, which is
validated by
the modeling results, is that the baseband transconductors dominate
the nonlinear
characteristics of integrated quadrature modulators. This motivates
design and modeling
work for characterization of the nonlinear distortion of a special
category of
transconductors: bipolar multi-tanh transconductors. Three baseband
and bandpass bipolar
transconductors: the bipolar differential pair; the multi-tanh
doublet; and the multi-tanh
triplet; are designed and their nonlinear characteristics are
modeled using the same
modeling technique as used with the quadrature modulator modeling
work: the AM-AM
and AM-PM based complex power series model. The bandwidth
limitations of the
AM-AM and AM-PM based model for characterizing baseband
transconductors are
studied and measured and an augmented model structure is proposed
to overcome the
limitations for broadband quadrature modulator applications.
Nonlinear Behavioral Modeling of Quadrature Modulators and Analysis
of Impacts on Wireless Communication Systems
by
Minsheng Li
A dissertation submitted to the Graduate Faculty of North Carolina
State University
in partial fulfillment of the Requirements for the Degree of
Doctor of Philosophy
APPROVED BY:
__________________________________
__________________________________ Dr. Kevin G. Gard Dr. Michael B.
Steer Chair of Advisory Committee
__________________________________
__________________________________ Dr. Huaiyu Dai Dr. Mark
Johnson
ii
Dedication
iii
Biography
Minsheng Li was born on January 13, 1977 in Jilin Province, P. R.
China. He received
his bachelor’s degree in 2000 from Tsinghua University, Beijing,
China.
Since 2002, Mr. Li has been studying in the Electrical and Computer
Engineering
Department at North Carolina State University, where he received
his Master of Science
degree in Electrical Engineering in 2003. Thereafter, he started
his PhD study under the
guidance of Dr. Kevin Gard with a focus on behavioral modeling of
microwave I/Q
quadrature modulators and radio frequency integrated circuit
design. He has published
major transactions and major conference publications.
He will be joining RF Micro Devices Scotts Valley Design Center in
the summer of
2007 as an analog/RF design engineer.
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Acknowledgements
First and foremost, I would like to express my deepest gratitude to
my advisor, Dr.
Kevin G. Gard, for his support, guidance, and encouragement
throughout my PhD study.
His wisdom, vision, and experiences have led me through my graduate
research career. I
would also like to thank Dr. Michael B. Steer, Dr. Huaiyu Dai, and
Dr. Mark Johnson for
their supports as my PhD committee members. Special thanks to Dr.
Michael B. Steer for
his insightful advice and discussions on many occasions.
I would like to extend a sincere thank you to Mr. Lowell Hoover
from Polyphase
Microwave Inc., for his supports in the quadrature modulator
modeling work.
I am also grateful to Dr. Steve Lipa, whose expertise in lab
measurement techniques
like on-wafer probing benefited me a lot in enhancing my lab
skills.
Thanks are also due to my good friends, Yingqun Yu, Hongyuan Zhang,
and Tao Jia
for their help in modeling and analysis of wireless communication
systems.
I would like to thank all the previous and current members of RAD
research group,
Xuemin Yang, Jie Hu, Gautham Krishnamurthy, Anand Vasani,
Shanthamurthy Prem
Swaroop, Jonathan Wilkerson, Anosh Davierwalla, David Mann, and
Sangtae Bae for
sharing their ideas and collaborating with me during my
dissertation research.
Last but not the least, I appreciate the support and understanding
from my fiancée, my
family and my friends.
1.1
Motivations.......................................................................................................................
1 1.2 Summary of Research Contributions
.................................................................................
2 1.3 Publications
......................................................................................................................
5 1.4 Dissertation
Organization..................................................................................................
6
2 Nonlinear Modeling of Quadrature Modulator Physical Impairments
.................................. 7 2.1 Overview of Quadrature
Modulator Applications in Wireless Communication Systems .....
7
2.1.1 Generic Quadrature Modulation Systems
...................................................................10
2.1.2 Zero-IF Single Sideband (SSB) OFDM Systems
........................................................12 2.1.3
Low-IF Single Sideband (SSB) OFDM Systems
........................................................14
2.2 Analog Impairments of I/Q Quadrature
Modulators..........................................................16
2.2.1 DC Offset (Carrier Leakage)
......................................................................................16
2.2.2 Gain and Phase
Imbalances........................................................................................18
2.2.3 Nonlinear Distortion
..................................................................................................22
2.4 Bandpass Nonlinearity Analysis
.......................................................................................32
2.4.1 Bandpass Nonlinearity for Power
Amplifiers..............................................................33
2.4.2 Bandpass Nonlinearity for Quadrature
Modulators.....................................................34
3.3 Gain/Phase Imbalances
....................................................................................................52
5 Design and Modeling of Bipolar Multi-Tanh
Transconductors..........................................117 5.1
Basics of
Multi-Tanh......................................................................................................117
5.2 Multi-Tanh Transconductors
Design...............................................................................120
5.3 Measurement Setup for Characterizing Multi-Tanh Transconductors
..............................128
5.4 Measurement and Simulation Results
.............................................................................139
5.4.1 Baseband
Transconductors.......................................................................................139
5.4.2 Bandpass Transconductors
.......................................................................................159
6 Conclusions and Future Work
..........................................................................................170
6.1 Summary of Research
...................................................................................................170
6.2 Future
Work...................................................................................................................173
Appendix B Design of a RF Integrated Direct-Conversion Quadrature
Modulator ............203
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List of Figures
Figure 2-1 Block diagram of a typical direct conversion
transmitter. .................................... 9 Figure 2-2
Spectrum of a RF modulated signal in a generic quadrature
modulation
system.
.............................................................................................................12
Figure 2-3 Spectrum of a RF modulated signal in a zero-IF SSB OFDM
modulation
(b) phase
error...................................................................................................19
Figure 2-7 Single sideband image rejection.
.......................................................................20
Figure 2-8 Image rejection characteristics for a zero-IF OFDM
system. ..............................21 Figure 2-9 Image rejection
characteristics for a low-IF OFDM
system................................22 Figure 2-10 Illustration
of nonlinear distortion: (a) example input and output
spectrum
in a nonlinear system; (b) zoomed view of the output spectrum.
.......................23 Figure 2-11 WLAN transmit spectrum mask
[32].
..............................................................27
Figure 2-12 Block diagram of the bandpass nonlinearity of a power
amplifier.....................33 Figure 2-13 Block diagram of the
bandpass nonlinearity of a quadrature modulator. ...........35
Figure 2-14 Power spectrums of IS-95 reverse-link signal.
.................................................39 Figure 2-15 RMS
percentage error vs. normalized truncation bandwidth for IS-95
signal....40 Figure 2-16 Power spectrums: (a) IS-95 reverse-link
signal; (b) the equivalent multisine. ...41 Figure 2-17 Gain
compression characteristic of the PA for IS-95 and multisine
excitations.
.......................................................................................................43
Figure 2-18 Comparison of ACPR for IS-95 and multisine excitations.
...............................44 Figure 2-19 Comparison of EVM for
IS-95 and multisine excitations. ................................45
Figure 3-1 Measured and calculated EVM versus carrier leakage for a
WCDMA
uplink signal.
....................................................................................................51
Figure 3-2 Measured carrier leakage in zero-IF WLAN and low-IF WLAN
systems. ..........52 Figure 3-3 Diagram of quadrature modulator
with gain and phase imbalance. .....................54 Figure 3-4
Measured and Calculated EVM contours versus gain/phase error
for
WCDMA and zero-IF WLAN signals.
..............................................................58
Figure 3-5 Measured spectrum plots for a low-IF WLAN signal: (1)
without
channel nonlinearities at baseband.
...................................................................64
Figure 3-9 Illustration of 3rd order distortion terms from the
four-tone analysis for
low-IF OFDM signals.
......................................................................................68
Figure 3-11 Spectrum plot of a quadrature modulator output of a
zero-IF WLAN signal. ....69 Figure 3-12 Spectrum plot of a
quadrature modulator output of a low-IF WLAN signal. .....71 Figure
3-13 Spectrum plot of a quadrature modulator output of a WCDMA
signal. .............74 Figure 3-14 Measured EVM of quadrature
modulator output excited with: 1) WCDMA
uplink signal; 2) WCDMA downlink signal; 3) zero-IF WLAN 16-QAM
signal; 4) low-IF WLAN 16-QAM signal.
.........................................................76
Figure 3-15 Measured upper and lower ACPR of quadrature modulator
output excited with zero-IF WLAN 16-QAM signal and low-IF WLAN
16-QAM signal. ........77
Figure 4-1 Block diagram of QM3337A quadrature modulator, after
[79]. ..........................83 Figure 4-2 Baseband equivalent
nonlinear model for QM3337A quadrature modulator. ......85 Figure
4-3 Measurement setup for AM-AM and AM-PM characterization
and
ACLR/EVM measurements.
.............................................................................90
Figure 4-4 Measured and predicted AM-AM and AM-PM characteristics
of I channel. .......91 Figure 4-5 Measured and predicted AM-AM and
AM-PM characteristics of Q channel.......92 Figure 4-6 Measurement
setup for four-point VNA measurements.
.....................................92 Figure 4-7 IM3 and ACLR
measurements over RF frequency.
.............................................95 Figure 4-8 Measured
and predicted ACLR (upper sideband) excited with the WCDMA
uplink pilot signal.
..........................................................................................101
Figure 4-9 Measured and predicted ACLR (lower sideband) excited
with the WCDMA
uplink pilot signal.
..........................................................................................101
Figure 4-10 Measured and predicted ACLR for (1) WCDMA uplink
signal;
Figure 4-12 Upper sideband and lower sideband ACPR for a 64 QAM
low-IF WLAN signal.
................................................................................................103
Figure 4-13 Difference between the measured and modeled ACPR for
low-IF WLAN signals.
...............................................................................................104
Figure 4-14 Measured and simulated EVM for 1) an uplink WCDMA
signal; 2) a downlink test model 5 WCDMA signal.
...................................................105
Figure 4-15 1) Upper plot: EVM results for a 64 QAM zero-IF WLAN
signal; 2) Lower plot: difference between the measured and
simulated EVM for the four zero-IF WLAN signals ( ( )EVMRCE log20=
)................................106
Figure 4-16 1) Upper plot: EVM results for a 64 QAM low-IF WLAN
signal; 2) Lower plot: difference between the measured and
simulated EVM for the four low-IF WLAN signals ( ( )EVMRCE log20=
). ...............................106
Figure 4-17 Measured and modeled image rejection and carrier
leakage over frequency....107 Figure 4-18 Diagram of the HMC496LP3
quadrature modulator (courtesy by
x
modulator excited by a 64-QAM zero-IF WLAN signal.
.................................113 Figure 4-22 Measured and
modeled ACPR (upper sideband).
...........................................114 Figure 4-23 Measured
and modeled ACPR (lower sideband).
...........................................114 Figure 4-24 Measured
and predicted EVM.
......................................................................115
Figure 5-1 Structure of a multi-tanh system.
.....................................................................119
Figure 5-2 Block diagram of a BJT differential pair
transconductor...................................120 Figure 5-3
Block diagram of a BJT multi-tanh doublet transconductor.
.............................121 Figure 5-4 Block diagram of a BJT
multi-tanh triplet transconductor.
...............................122 Figure 5-5 Analytical
single-tone gain compression curves of three baseband
transconductors...............................................................................................124
Figure 5-6 Simulated IM3 over input voltage for the three baseband
transconductors. .......125 Figure 5-7 Simulated Gm of the
differential pair, doublet, and triplet transconductors. .......126
Figure 5-8 Simulated IM3 over input power for the three bandpass
transconductors. .........127 Figure 5-9 Measurement setup with
network analyzer only.
..............................................129 Figure 5-10
Measurement setup with network analyzer and baluns.
..................................130 Figure 5-11 Simulated HD3,
HD5, HD7 with 2 dB and without amplitude imbalance. ........134
Figure 5-12 Simulated HD3, HD5, HD7 with 10 degree and without
phase imbalance. ......138 Figure 5-13 Small signal gain-bandwidth
plot of baseband differential pair
transconductor.
...............................................................................................141
Figure 5-14 Small signal gain-bandwidth plot of baseband doublet
transconductor............142 Figure 5-15 Small signal
gain-bandwidth plot of baseband triplet transconductor.
.............142 Figure 5-16 Measured, simulated, and analytical
gain characteristics of baseband
differential pair
transconductor........................................................................144
Figure 5-17 Measured, simulated, and analytical gain
characteristics of baseband
transconductor. excited by WCDMA downlink test model 5 signal.
.................151
xi
Figure 5-25 Measured and modeled ACLR for baseband multi-tanh
doublet transconductor excited by WCDMA downlink test model 5
signal...................152
Figure 5-26 Measured and modeled ACLR for baseband multi-tanh
triplet transconductor excited by WCDMA uplink pilot signal.
..................................153
Figure 5-27 Measured and modeled ACLR for baseband multi-tanh
triplet transconductor. excited by WCDMA downlink test model 5
signal. .................153
Figure 5-28 Simulated AM-AM and AM-PM of baseband differential pair
transconductor over frequency.
.......................................................................155
Figure 5-29 Measured and simulated AM-PM of baseband differential
pair transconductor over frequency.
.......................................................................156
Figure 5-30 Measured and simulated AM-AM of baseband differential
pair transconductor over frequency.
.......................................................................156
Figure 5-31 Measured and simulated AM-PM of baseband multi-tanh
doublet transconductor over frequency.
.......................................................................157
Figure 5-32 Measured and simulated AM-AM of baseband multi-tanh
doublet transconductor over frequency.
.......................................................................157
Figure 5-33 Measured and simulated AM-PM of baseband multi-tanh
triplet transconductor over frequency.
.......................................................................158
Figure 5-34 Measured and simulated AM-AM of baseband multi-tanh
triplet transconductor over frequency.
.......................................................................158
Figure 5-35 Measured, simulated, and analytical gain
characteristics of the bandpass differential pair
transconductor.........................................................160
Figure 5-36 Measured, simulated, and analytical gain
characteristics of the bandpass multi-tanh triplet transconductor.
.....................................................161
Figure 5-37 Measured, modeled, simulated, and analytical IM3 of the
bandpass differential pair
transconductor........................................................................162
Figure 5-38 Measured, modeled, simulated, and analytical IM3 of the
bandpass multi-tanh triplet transconductor.
....................................................................163
Figure 5-39 Simulated and measured ACLR for the bandpass
differential pair transconductor excited by the WCDMA uplink pilot
signal. ............................164
Figure 5-40 Simulated and measured ACLR for the bandpass
differential pair transconductor excited by the WCDMA downlink test
model 5 signal. ............165
Figure 5-41 Simulated and measured ACLR for the bandpass multi-tanh
triplet transconductor excited by the WCDMA uplink pilot signal.
............................165
Figure 5-42 Simulated and measured ACLR for the bandpass multi-tanh
triplet transconductor excited by the WCDMA downlink test model 5
signal. ............166
Figure 6-1 A simple augmented AM-AM and AM-PM based two-box model.
...................174 Figure B-1 Top-level diagram of the
quadrature modulator.
..............................................209 Figure B-2 Gm
plot: (a) for CMOS triplet transconductor (b) for CMOS
triplet
transconductor and differential pair transconductor.
........................................213 Figure B-3 Simulated
IM3 for (a) quadrature modulator w/ differential pair
transconductor
xii
(b) cascade stage w/ CMOS triplet
transconductor...........................................217 Figure
B-5 Spot noise for (a) quadrature modulator w/ differential pair
transconductor
and (b) quadrature modulator w/ CMOS triplet transconductor.
.......................220 Figure B-6 Image suppression: (a)
quadrature modulator w/ differential pair
transconductor and (b) quadrature modulator w/ CMOS triplet
transconductor.
...............................................................................................222
xiii
xiv
AM-AM Amplitude modulation to amplitude modulation
AM-PM Amplitude modulation to phase modulation
AWGN Additive White Gaussian Noise
BER Bit Error Rate
DFT Discrete Fourier Transform
DUT Device Under Test
EVM Error Vector Magnitude
FFT Fast Fourier Transform
IRR Image Rejection Ratio
P1dB 1 dB gain compression point
PA Power amplifier
xv
SNDR Signal to Noise plus Distortion Ratio
SNR Signal to Noise Ratio
SSB Modulation Single Sideband Modulation
VNA Vector Network Analyzer
VSA Vector Signal Analyzer
1
The quadrature modulator is an essential component in modern
wireless
communication systems because of their high spectrum efficiency. It
is used in different
manners in different applications including cellular phones,
wireless data communication
systems, Bluetooth, GPS, and etc. In order to efficiently design
and optimize quadrature
modulators, one concern is to understand how the quadrature
modulators are used in these
different applications, and how the physical impairments such as
nonlinear distortion, the
DC offset, and gain/phase imbalances impact the system performance
in different wireless
systems. Judicious design decisions can be made only after gaining
such knowledge in
order to achieve optimized circuit and system designs.
Another concern in quadrature modulator design is finding a way to
quickly and
accurately characterize the quadrature modulator physical
impairments for system design
and verification. System-level behavioral modeling techniques are
desired because they
2
can simulate faster with less memory demands than circuit models.
Recent literature
addressed the modeling of the electrical characteristics of mixers
at the circuit level [1–4]
but very little work has been done for system-level nonlinear
behavioral modeling of
quadrature modulators. However, a considerable amount of work has
been done for
modeling the nonlinear characteristics of power amplifiers [5–14].
Because the nonlinear
distortion in a quadrature modulator can cause significant
performance degradation in
terms of spectral growth and waveform quality, it is desirable to
implement the nonlinear
behavioral modeling techniques for power amplifiers to quadrature
modulators for
characterization the nonlinear distortion and other physical
imperfections.
1.2 Summary of Research Contributions
Prevailing usage of quadrature modulators are represented by the
following three
systems: 1) a generic quadrature modulation system; 2) a zero-IF
OFDM system; and 3) a
low-IF OFDM system. Interestingly, the physical impairments of the
quadrature
modulators in different applications impact the system performance
in different manners as
well. Extensive analysis work was done in Chapter 3 to reveal the
underlying mechanisms
that result in the various impacts of the quadrature modulator
physical imperfections in
different applications. From these system and circuit designers
gain a better understanding
about the design tradeoffs, and wisely diagnose the physical
imperfections appropriate for
different quadrature modulator applications to improve their design
and performance. The
impact of the nonlinear distortion, of DC offset, and of gain/phase
imbalances on
quadrature modulator performance degradation in terms of spectral
regrowth and
waveform quality degradation are summarized in Table 3-4.
3
In this work, to the best knowledge of the author, for the first
time an accurate
nonlinear behavioral model is developed for characterization of the
correlated/uncorrelated
nonlinear distortion and of the linear static errors. These include
the DC offset and the
gain/phase imbalances of the direct-conversion quadrature
modulators. Using this model
the modulator performance degradations can be predicted
efficiently. The baseband
equivalent complex power series model structure is adopted because
it is easy to
implement without loss of accuracy for quadrature modulator
applications. A novel
AM-AM and AM-PM measurement technique is developed to extract the
nonlinear model
parameters. Earlier approach used a swept DC input, but here a
swept offset single-tone
sinusoidal input in the measurement is used so that the carrier
leakage induced dynamic
range reduction problem is overcome. For the extraction of the
linear error model
parameters, an accurate four-point VNA measurement technique is
developed that is
conveniently performed as only a DC source and a VNA are needed. An
orthogonalization
technique developed previously [15, 16] for power amplifiers was
extended to quadrature
modulator applications in this work so that the correlated and
uncorrelated nonlinear
distortion can be decomposed and thus the waveform quality
degradation can be accurately
predicted by the model. The details of model development,
parameters extraction
procedures, and model verifications are presented in Sections 4.1
and 4.2.
In integrated direct-conversion quadrature modulators, usually the
baseband
transconductors are the dominant nonlinear components, therefore,
the nonlinear
characteristics of the individual baseband transconductors are
investigated to gain more
insights regarding nonlinear quadrature modulator modeling. In
Chapter 5, the design of a
4
modeled for characterization of its nonlinear responses. The same
nonlinear model
structure as used in Chapter 4, a complex power series model
structure based on the
measured AM-AM and AM-PM data, is utilized for characterizing the
nonlinear responses
of the baseband and bandpass transconductors. The bandwidth
limitations due to the
input/output parasitics are analyzed and measured for the complex
power series model of
the baseband transconductors, based on which an augmented two-box
model structure is
proposed for broadband quadrature modulator applications.
The final contribution is partly related to the nonlinear
characterization work which is
about the modeling of multisine signals. Multisine signals have
gained more and more
attentions in behavioral modeling, testing, and characterization of
nonlinear devices
because they can mimic real communication signals to capture
broadband nonlinear
characteristics and characterize both in-band and out-of-band
distortion. In Section 2.5, a
novel FFT based technique for multisine signal generation was
presented. Compared to
other multisine design techniques, our approach is straightforward
and easy to implement.
In summary, these contributions are listed below in the order they
appear:
• The development of a novel FFT based technique for generating
multisine signal, one
particular type of signal for ease of nonlinear characterization
(Section 2.5).
• System-level analysis of the impacts of quadrature modulator
physical impairments
on wireless systems in three different modulator applications: 1)
generic quadrature
modulations system, 2) zero-IF SSB OFDM system, and 3) low-IF SSB
OFDM
5
system. The impacts of the DC offset, the gain/phase imbalances,
and the nonlinear
distortion are analyzed in Sections 3.2, 3.3 and 3.4,
respectively.
• Behavioral model development and novel parameter extraction
procedures for both
passive (Section 4.1) and active (Section 4.2) direct-conversion
quadrature
modulators for characterization of nonlinear distortion and linear
static errors.
• Design and modeling of baseband (Sections 5.2 and 5.4.1) and
bandpass (Sections
5.2 and 5.4.2) transconductors for characterization of the
nonlinear distortion. The
bandwidth limitations of complex power series models for
characterizing the
baseband transconductors were analyzed and measured (Section
5.4.2). The
limitations of the balun-based differential probing measurement
technique were
analyzed for nonlinear differential on-wafer probing
characterization (Section 5.3).
1.3 Publications
Current Publications:
M. Li, L. Hoover, K. G. Gard and M. B. Steer, “Behavioral Modeling
and Impact Analysis
of Physical Impairments of Quadrature Modulators”, Submitted for
review to IEEE
Transactions on Microwave Theory and Techniques, 2007.
M. Li, L. Hoover, K. G. Gard and M. B. Steer, “Behavioral Modeling
for I/Q Quadrature
Modulators for Characterization of Nonlinear Distortion”, IEEE
MTT-S International
Microwave Symposium, June 2006.
6
M. Li, K. M. Gharaibeh, K. G. Gard and M. B. Steer, “Accurate
Multisine Representation
of Digital Communication Signals for Characterization of Nonlinear
Circuits”, IEEE
Radio and Wireless Symposium, Jan. 2006.
Future Publications:
M. Li, K. G. Gard and M. B. Steer, “Design and Behavioral Modeling
of Multi-Tanh
Transconductors for Characterization of Nonlinear Distortion”, in
preparation.
1.4 Dissertation Organization
Chapter 2 reviews the three different quadrature modulator
applications in wireless
communication systems, the background knowledge about the physical
impairments of
quadrature modulators, and the fundamentals of nonlinear behavioral
modeling techniques,
all of which serve as the basis for the later chapters. The
multisine signal generating
technique is documented in Section 2.5. In Chapter 3, the impacts
of the quadrature
modulator physical impairments on wireless systems in three
different modulator
applications are discussed. The model development and parameter
extraction procedures
for passive and active direct-conversion quadrature modulators are
elaborated in Chapter 4.
The design and modeling of the bipolar transconductors are
presented in Chapter 5.
Conclusions and future areas of work are presented in Chapter 6. As
an extension to the
transconductor design and modeling work in Chapter 5, a CMOS RF
integrated
direct-conversion quadrature modulator is designed and simulation
results are presented in
Appendix B.
Impairments
Communication Systems
Since 1901 Guglielmo Marconi successfully transmitted radio signals
across the
Atlantic Ocean, wireless technology has undergone an incredible
revolution, thanks to the
inventions of the vacuum tubes [17, 18], the transistors [19], the
integrated circuits [20],
and the developments of Shannon’s information theory [21] and the
cellular system
conception. Affordable mobile communications in the areas such as
cellular phones,
Bluetooth, wireless data communications, global positioning system
(GPS) have greatly
facilitated human’s daily life today. For example, the Cellular
Telecommunications &
Internet Association (CTIA) reports that 233,040,781 wireless
subscribers exist in US as of
the end of 2006 [22]. This huge market in wireless communications
drives the industry to
8
provide competitive services and equipments to raise the market
share. A key factor of
success is the provision of high performance transceivers with
lower cost, lower power
consumption, smaller size, higher reliability and more
functionality, which pushes the
designers to design the transceivers closely approaching the system
requirements in order
to maximize the yield.
Two types of transceiver architectures are prevailed in modern
wireless systems: the
super-heterodyne architecture [23–25] and the homodyne, also known
as direct conversion
architecture [24–26]. In this thesis, the focus is on the modeling
of the direct conversion
transmitters.
The direct conversion principle was first developed in 1932 as an
attempt to surpass
the super-heterodyne [26]. It was not widely used for a long time
because it has a number
of issues including DC offset, I/Q imbalances, and flicker noises.
However, it has been
resurrected in recent years due to its advantages of adaptability
to different standards, small
form factor, low cost, reduced bill of materials and low power
consumption over
super-heterodyne architecture [27]. Also, the improved
semiconductor technology and
system level optimization of the circuits make it possible to
alleviate the above problems.
As a good guidance to the designers, in [28], Abidi summarized the
key problems of using
direct-conversion transceivers and the various solutions to them.
As shown in Figure 2-1, a
typical direct conversion transmitter consists of a number of
concatenated building blocks
such as baseband processors, digital-to-analog converters (DAC),
variable gain amplifiers
(VGA), quadrature modulator, driver amplifier (DA), and surface
acoustic wave (SAW)
filters.
9
Among the many components in the wireless transmitters (both
super-heterodyne and
direct conversion), the quadrature modulator plays an important
role by conducting both
modulation and frequency conversion functions. The blue dash line
enclosed block in
Figure 2-1 is the simplified diagram of a direct-conversion
quadrature modulator. The local
oscillator (LO) input, at the carrier frequency, is split into both
inphase and 90o quadrature
phase signals which are fed to two balanced mixers. The baseband
input signals are
subdivided into two independent data streams, i(t) and q(t), and
mixed with the in-phase
and quadrature carrier signals separately and combined in-phase to
generate linear
quadrature modulation of the RF carrier. The major advantage of
using the quadrature
modulators is the increase of the spectrum efficiency because the
i(t) and q(t) data streams
can be modulated onto the same carriers orthogonally so that
different quadrature
modulation schemes such as quadrature amplitude modulation,
quadrature phase
modulation, or theirs combinations can be achieved.
Receiver
DAC
∑∑ PA
DAC
Figure 2-1 Block diagram of a typical direct conversion
transmitter.
One important property of the I/Q quadrature modulator is to
perform single-sideband
10
(SSB) modulation using phasing to suppress the unwanted sideband.
If i(t) and q(t) are 90
degree out of phase, the resulting modulator output signal is
single sideband [29]. This
property can be illustrated by a simple example below. Let
)sin()( );cos()( ttqtti BBBB ωω == . (2.1)
where BBω is the baseband frequency. The modulator output is:
)sin()sin()cos()cos()sin()()cos()()( ttttttqttitw cBBcBBcc ωωωωωω
−=−= . (2.2)
where cω is the carrier frequency. By trigonometry,
[ ]ttw BBc )(cos)( ωω += . (2.3)
The quadrature modulator output in Equation (2.3) is a single
sideband result. In summary,
if the input I and Q signals are 90 degree out of phase, after
quadrature mixing with the
carrier signals that are 90o out of phase and then summing
together, a lower or upper
sideband signal results.
Due to the ability of conserving bandwidth, the quadrature
modulators have become
very popular in modern wireless systems where high data bandwidth
is a critical
performance metric. The use of quadrature modulators can be
categorized into three types
of applications depending on the transceiver design and common
practice for meeting
specifications: 1) generic quadrature modulation systems; 2)
zero-IF SSB OFDM systems;
and 3) low-IF SSB OFDM systems.
2.1.1 Generic Quadrature Modulation Systems
The generic quadrature modulation systems are commonly used for
cellular
applications such as GSM/EDGE, CDMA, and WCDMA [25]. In these
cellular systems,
11
the baseband information is subdivided and mapped to independent
i(t) and q(t) data
streams as shown in Figure 2-1 depending on different modulation
schemes being used.
The I and Q data streams are then upconverted to a single carrier
with the quadrature
modulators by means of double sideband (DSB) modulation. The
carrier signal with
amplitude and phase modulation can be represented as:
[ ]tj cetCtw ω)(~)( ℜ= . (2.4)
where )(~ tC is the carrier information which is referred to as
complex envelope.
)()()()(~ )( tjqtietAtC tj +== θ . (2.5)
[ ] )sin()()cos()()(cos)()( ttqttitttAtw ccc ωωθω −=+= .
(2.6)
Viewed in the frequency domain as shown in Figure 2-2, the baseband
modulation
information of I and Q channel is upconverted to the single carrier
cω and overlapped on
top of each other orthogonally so that they can be separated and
detected at the receiver,
which means that using quadrature modulation same amount of RF
spectrum can carriers
double amount of information (I and Q) and this is the reason why
quadrature modulators
are able to improve spectrum efficiency.
12
ωc
I data Q data
Figure 2-2 Spectrum of a RF modulated signal in a generic
quadrature modulation system.
2.1.2 Zero-IF Single Sideband (SSB) OFDM Systems
The zero-IF OFDM systems are popularly used for wireless data
communication
applications such as WLAN and WiMAX. The orthogonal frequency
division multiplexing
(OFDM) is a type of multi-carrier modulation technique, which
splits the transmit
bandwidth into multiple orthogonal sub-channels that are narrow
enough and thus
experience a flat fading although the overall radio propagation
environment is
frequency-selective. The spectra orthogonality is achieved by
carefully selecting the
carrier spacing so that each sub-carrier is located on all the
other sub-carriers’ spectra zero
crossing points, as shown in Figure 2-3. Each sub-carrier carries
unique narrow-band
modulated information in parallel and very high data rate can be
obtained with many
sub-carriers in a system. The OFDM systems can effectively tackle
the inter symbol
interference (ISI) problem and allow to achieve high data rate in a
frequency-selective
radio propagation environment [30, 31]. The OFDM modulation
technique is proposed as
the air-interface solution for wireless local area networks (WLANs)
[32], wireless
13
metropolitan area networks (WMAN) [33], and possibly for the future
fourth-generation
mobile cellular wireless systems [34].
ωcωc
Figure 2-3 Spectrum of a RF modulated signal in a zero-IF SSB OFDM
modulation system.
One implementation of the OFDM transceiver is the zero-IF
architecture [35], where
all the sub-carriers are located symmetrically around the carrier
frequency, as presented in
Figure 2-3. The I/Q inputs of a zero-IF OFDM signal to the
quadrature modulator are
presented in Equation (2.7).
ketAtjQtItC θ=+= represents the location of the symbols
within
the constellation for the kth subcarrier at different symbol time,
i.e., the modulation
information transmitted by the kth sub-carrier. Notice that i(t)
and q(t) of a zero-IF OFDM
signal are Hilbert transform related so that each sub-carrier is
upconverted to the carrier
14
frequency by single-sideband modulation. Different from the generic
quadrature
modulation system, the zero-IF OFDM signals have to employ the
quadrature modulators’
single-sideband modulation property to upconvert each sub-carrier.
The RF zero-IF OFDM
modulated signal is:
tj k tttAeetCtw ck )()(cos)()()( θωωωω . (2.8)
Equation (2.8) shows that each sub-carrier is modulated by the
unique information
)( and )( ttA kk θ and upconverted to single-sideband )( kc ωω + .
The spectrum of a zero-IF
OFDM signal represented by Equation (2.8) is shown in Figure 2-3.
Note that in a real
OFDM system, the sub-carrier at the carrier frequency is usually
not used to avoid carrier
leakage problem, which will be discussed in Section 1.2.1.
2.1.3 Low-IF Single Sideband (SSB) OFDM Systems
Another implementation of the OFDM transceiver is the low-IF
architecture [36, 37],
where all the sub-carriers are located symmetrically around the LO
plus an IF frequency
)( IFLO ωω + in stead of the carrier frequency, as shown in Figure
2-4. The I/Q inputs of a
low-IF OFDM signal to the quadrature modulator are shown in
Equation (2.9). Compared
[ ] [ ]{ }
[ ] [ ]{ }∑∑
∑∑
−=−=
−=−=
+++=
ketAtjQtItC θ=+= represents the location of the symbols
within
the constellation for the kth subcarrier at different symbol time,
i.e., the modulation
15
( )[ ]∑∑ −=−=
+++=
tjtj k tttAeeetCtw cIFk )()(cos)()()( θωωωωωω . (2.10)
Equation (2.10) shows that each sub-carrier is modulated by the
unique information
)( and )( ttA kk θ and upconverted to single-sideband [ ])( IFkc
ωωω ++ . Similar as the
zero-IF OFDM signals, the low-IF OFDM signals have to utilize the
quadrature
modulators’ single-sideband modulation property to upconvert each
sub-carrier that carries
unique modulation information.
ωLO ωLO+ ωIF
Figure 2-4 Spectrum of a RF modulated signal in a low-IF SSB
modulation system.
Compared to the zero-IF OFDM architecture, the low-IF OFDM
architecture has the
advantages of reduced in-band distortion and no SNR degradation due
to the carrier
feedthrough and the quadrature gain and phase imbalances. The
drawbacks of the low-IF
are the more stringent requirement on the DACs because the sampling
rate has to be at least
doubled and the increased adjacent channel interferences. The
tradeoffs between the
16
zero-IF OFDM and low-IF OFDM architectures will be discussed in
detail in Chapter 4.
2.2 Analog Impairments of I/Q Quadrature Modulators
Imperfections in analog circuits such as device mismatches,
nonlinearity and noise
will cause degradation of I/Q quadrature modulator performance.
Such physical
impairments include DC offset, gain/phase imbalances, nonlinear
distortion, phase noise,
frequency error, spurious, and etc. The analog imperfections
produce error products which
result in degradation of signal-to-noise ratio (SNR) or
equivalently EVM and undesired
spectral occupancy of the transmitted signals in the wireless
systems. These quadrature
modulator impairment impacts are possible to be alleviated. For
example, the techniques
for estimating and compensating the DC offset and quadrature
imbalances were proposed
in [38, 39].
In this thesis, the DC offset, gain/phase imbalances, and nonlinear
distortion in a direct
conversion quadrature modulators were accurately modeled and the
effects on the signal
quality were analyzed.
2.2.1 DC Offset (Carrier Leakage)
DC offset in a quadrature modulator results from the device
mismatches such as
threshold voltage mismatch, device size mismatch, resistor
mismatch, and etc. It leads to
carrier feedthrough, which is a major concern in designing
direct-conversion transceivers.
In a quadrature modulator, the mismatch can be modeled as a DC
offset in the baseband
input signals, as shown in Equation (2.11).
[ ] [ ])()()(~ ,, tqVjtiVtC QOSIOS +++= . (2.11)
17
where VOS,I and VOS,Q are the DC offset voltage, i(t) and q(t) are
the desired I and Q
modulation data.
In a generic quadrature modulation system, the DC offset causes an
offset of the signal
constellation by the amount of VOS,I and VOS,Q [40] as shown in
Figure 2-5 an offset QPSK
constellation diagram.
( ) [ ] [ ])sin()()cos()()sin()cos()(~)( ,, ttqttitVtVetCtw
cccQOScIOS tj c ωωωωω −+−=ℜ= . (2.12)
As seen in Equation (2.12), the DC offset produces a carrier
leakage
term [ ] ,)sin()cos( ,, tVtV cQOScIOS ωω − which is uncorrelated to
the desired signal
[ ])sin()()cos()( ttqtti cc ωω − and causes the SNR degradation. In
the zero-IF and low-IF
OFDM systems, usually there is no sub-carrier at DC so that the
carrier leakage will not
cause SNR degradation. However, the carrier leakage is still
possible to cause violation of
18
2.2.2 Gain and Phase Imbalances
As shown in Figure 2-1, in the quadrature modulators, the LO output
needs to be
shifted by 90o to perform quadrature mixing. The analog impairments
can cause errors in
the 90o phase shift and mismatches of the amplitude of the I and Q
channels. These gain
and quadrature phase errors result in distortion of the transmitted
signal constellation and
ultimately the degradation of the bit error rate (BER). The effects
of the gain and
quadrature phase errors on the transmitted signal constellation can
be illustrated by a
QPSK example. Suppose the baseband signal i(t) and q(t) are:
btq ati
. (2.13)
where a and b are either +1 or -1. With the existence of the gain
and phase imbalances, let
−
−=
+
+=
ε
tV
tV
cQLO
cILO
. (2.14)
( ) ( )
−
−−
+
+=
Simplify (2.15) by trigonometry,
. (2.16)
By a close examination of (2.16), it can be found the equivalent
baseband signal í(t)
( ) ( )
( ) ( )
++
−=
−+
+=
bati . (2.17)
As shown in Equation (2.17), the baseband information was distorted
by the I/Q gain and
phase errors. Their effects on the transmitted QPSK signal
constellation were shown in
Figure 2-6: the gain error changes the signal constellation from
square to rectangular, and
the quadrature phase error causes skew of the constellation.
Q
I
Ideal
(a)
Q
I
Ideal
(b)
Q
I
Ideal
Q
I
IdealIdeal
(a)
Q
I
Ideal
(b)
Q
I
Ideal
(b)
Figure 2-6 Effects of I/Q imbalances on QPSK constellation: (a)
gain error, (b) phase error.
The I/Q gain and phase imbalances are usually characterized by the
single tone SSB
20
image rejection ratio (IRR) [25]. In such a test, the single tone
SSB baseband signals as
described in (2.1) were applied to the quadrature modulator with
gain and phase errors. The
resulting RF spectrum was shown in Figure 2-7. Besides the desired
upper sideband
( BBc ωω + ), there is an image spectrum generated at ( BBc ωω − ).
The IRR is defined as the
++ +−
θ θ . (2.18)
where θ is the quadrature phase error in degree and is the square
of the ratio of the
amplitude of I channel to that of Q channel.
ωc
Figure 2-7 Single sideband image rejection.
The quadrature errors in the generic quadrature modulation systems
can be seen as the
I/Q power leaking onto Q/I. This causes a "distortion floor" that
leads to the degradation of
the SNR or EVM of the signal. A generic quadrature modulation
system is single carrier
system. The gain and quadrature phase errors cause the distortion
of the signal
21
constellation as shown in Figure 2-6.
In a zero-IF OFDM modulation system, the results of the I/Q
imbalances can be seen
as imperfect sideband cancellation. With the existence of the
gain/phase imbalances, the kth
sub-carrier generates an image spectrum at the –kth sub-carrier, as
shown in Figure 2-8,
which is uncorrelated to the desired signal so that its impacts are
very similar to the
additive white Gaussian noise (AWGN) degrading the SNR and EVM of
the zero-IF
OFDM signals. Because the zero-IF OFDM system is a multi-carrier
system with each
sub-carrier carrying unique information, the I/Q imbalances cause
distortion of the
constellation of each sub-carrier and the total effect is
manifested as a spreading of the
constellation in a noise-like fashion [40].
Image Rejection
Figure 2-8 Image rejection characteristics for a zero-IF OFDM
system.
The advantage of a low-IF OFDM modulation system is that the I/Q
imbalances won’t
distort the transmitted signal constellation and cause SNR or EVM
degradation. This
doesn’t mean that no image spectra are generated from the I/Q
imbalances, but the image
22
spectra are shifted out of the signal bandwidth to the adjacent
sideband as shown in Figure
2-9 so that no SNR or EVM degradation occurs. However, it is
obvious that the signal in
the adjacent channel will suffer because now the image products are
the interferences to the
adjacent channel.
Figure 2-9 Image rejection characteristics for a low-IF OFDM
system.
2.2.3 Nonlinear Distortion
Nonlinear distortion in analog/RF circuit and system design poses a
challenging
problem to the designers. When an amplitude or phase modulated
signal passes through a
nonlinear system, new frequency components are usually generated in
the output, as shown
in Figure 2-10(a). The nonlinearities produce nonlinear distortion
both inside and outside
the signal band, as shown in Figure 2-10(b). The out-of-band
nonlinear distortion leaks into
the adjacent channels and causes SNR degradation of the signals in
the adjacent bands.
Therefore, in wireless system designs, stringent requirements are
specified for the allowed
maximum out-of-band emissions. Another form of nonlinear distortion
is generated by
23
means of cross modulation that transfers the modulation on the
amplitude of the interferer
to the amplitude of the desired signal. It is especially an
important problem in a
multi-channel system [41]. Also shown in Figure 2-10(b), the
in-band distortion can be
decomposed into two components: one is correlated with the desired
signal, which causes
gain compression or expansion; the other is uncorrelated with the
desired signal and
behaves like AWGN to degrade the effective system SNR or
equivalently the EVM of the
desired transmitted signals, ultimately leads to the degradation of
the BER.
Uncorrelated In-band Distortions
-10 -8 -6 -4 -2 0 2 4 6 8 10 -70
-65
-60
-55
-50
-45
-40
-35
-30
-25
Correlated In-band Distortions
Figure 2-10 Illustration of nonlinear distortion: (a) example input
and output spectrum in a nonlinear system; (b) zoomed view of the
output spectrum.
The effective system SNR is the ratio of the signal power to the
total noise plus the
uncorrelated in-band distortion power, which is usually referred to
as the signal to noise
(a)
(b)
24
and distortion power (SNDR) [42, 43]. It is a function of both the
nonlinear distortion and
the AWGN, as shown in Equation (2.19).
nod
oc
PP P +
=SNDR . (2.19)
where odP is the uncorrelated distortion power, nP is the power of
the AWGN, and ocP is
the power of the output components correlated to the input
signals.
The EVM is an important figure-of-merit for characterizing the
waveform quality for
digital modulated signals, which is a measure of the difference
between the reference
waveform and the actual waveform. It is defined as the square root
of the ratio of the mean
error vector power to the mean reference power expressed as a %.
EVM is directly related
to the system SNDR [43]:
%100 SNDR
1(%)EVM ×= . (2.20)
In OFDM systems, instead of using the term of EVM, Relative
Constellation Error (RCE)
is used as the figure-of-merit which is exactly equivalent to
EVM:
)EVMlog(20RCE(dB) = . (2.21)
SNDR(dB)RCE(dB) −= . (2.22)
so that RCE is a straightforward measure of SNDR of the OFDM
signals.
In the thesis, three classes of digital wireless signals, WCDMA,
zero-IF WLAN, and
low-IF WLAN signals were extensively used in system analysis and
model validation
work. The EVM/RCE specifications for these transmit signals were
summarized in Table
25
2-1 and 2-2.
Table 2-1 ACLR and EVM specifications in WCDMA systems [44,
45].
3GPP Specifications Parameters
User Equipment Base Station ACLR1 ( 5± MHz) (dBc) 33 45 ACLR2 ( 10±
MHz) (dBc) 43 50 EVM (%) 17.5 17.5*, 12.5**
* EVM when base station is transmitting a composite signal using
only QPSK modulation. ** EVM when the base station is transmitting
a composite signal that includes 16QAM modulation.
Table 2-2 EVM specifications for WLAN transmitted signals (802.11g)
[32].
Data Rate (Mbps) RCE (dB) Equivalent EVM (%) 6 -5 56.2 9 -8
39.8
12 -10 31.6 18 -13 22.4 24 -16 15.8 36 -19 11.2 48 -22 7.9 54 -25
5.6
Modern wireless communications systems such as GSM/EDGE, CDMA,
WCDMA,
WLAN and WiMAX are multi-channel systems where the RF spectrum is
split into
multiple channels to accommodate multiple users [46]. Out-of-band
emissions from the
transmitter leaking into adjacent channels can degrade the SNR of
the signals in those
channels. Therefore, stringent requirements are specified in
various digital wireless
standards for maximum allowable out-of-band emissions from the
transmitter. ACPR and
spectrum mask are two important factors for this use.
26
As shown in Figure 2-10(b), the spectrum mask specify a set of
limiting value on the
transmitted signal spectra over frequency and no spectrum can go
beyond the mask. ACPR
is a key figure-of-merit in quantizing the adjacent channel
interference, which is defined as
the ratio (in decibel) of the distortion power in a certain
bandwidth with a certain frequency
offset from the carrier frequency, and the power in the desired
channel with a certain
∫ ∫
⋅
⋅ =
dffS . (2.23)
where f1 and f2 are the lower and upper frequency limit of the main
channel; f3 and f4 are the
lower and upper frequency limit of the adjacent channel. Each
wireless standard has its
own specification values for the definition of ACPR in order to
evaluate the impacts of the
out-of-band interferences on the SNR degradation of the user
signals in the adjacent
channel. Note that WCDMA systems use a different term name,
adjacent channel leakage
ratio (ACLR), which is exactly the same as ACPR. In WLAN systems,
there is no ACPR
definition because the EVM is a more stringent system
specification.
The ACLR specifications for WCDMA transmit signals were listed in
Table 2-1. The
spectrum mask for WLAN transmit signals is shown in Figure
2-11.
)sin()cos( ,, tVtV cQOScIOS ωω −
Figure 2-11 WLAN transmit spectrum mask [32].
In [15, 16], the authors successfully developed an orthogonal
behavioral model to
decompose the correlated and uncorrelated components of the output
signal from RF
power amplifiers so that the system SNR or EVM degradation can be
estimated. In this
thesis, this orthogonal behavioral modeling technique for power
amplifiers is extended to
the direct-conversion quadrature modulators for uncorrelated and
correlated in-band
distortion decomposition.
2.3 Behavioral Modeling Techniques for Nonlinear Circuits
As a compact representation of a circuit or a system, a behavioral
model typically
simulates much faster and requires less memory than its
circuit-level counterpart, which
makes it a powerful tool for system design and verification.
Behavioral modeling is a
black-box modeling approach to characterize a device since it can
be generated without
knowing the detailed circuit structure. This property also leads to
another advantage of a
behavioral model that the vendors can supply the customers only a
black-box model
28
without revealing the detail information of a device to protect the
intellectual property [47,
48].
A behavioral model is usually developed based on either simulation
or measurement
data, which are a set of wisely selected input-output observations.
Compared to simulation
based model development, a measurement based model development can
capture actual
nonlinear characteristics of a device since it is not affected by
the inaccuracy of the
underlying circuit models. Obviously, the accuracy of the
behavioral models heavily
depends on the adopted model structure and model parameter
extraction procedure [47,
49].
In this thesis, a measurement based behavioral model for a
direct-conversion
quadrature modulator was developed based on the previous modeling
techniques for power
amplifiers.
2.3.1 Behavioral Modeling for Power Amplifiers
Design of the power amplifier in RF systems is almost the most
challenging task due
to its nonlinear impairments and high power output/efficiency
requirement. A lot of work
has been done in the area of behavioral modeling for power
amplifiers in order to
accurately characterize the nonlinear behavior in a PA [6, 7, 9–14,
50–54]. Most PA models
are based on bandpass nonlinearity concepts that only the output
signal envelope responses
around the carrier frequency were captured known that only the
signal envelope carries the
useful information and the final PA outputs are bandpass filtered.
It differs from the
instantaneous PA models that deal with the complete RF signal
including all the nonlinear
effects and harmonics. The bandpass nonlinearity based PA models
greatly ease the
29
simulation in terms of simulation time and memory requirements by
ignoring the carrier
effects and the higher order harmonics.
The simplest but very popular PA behavioral model for modeling the
nonlinear
distortion is the memoryless model. The commonly used baseband
equivalent memoryless
models are in the form [55]:
[ ] [ ]kK
k
+ += . (2.24)
where )(~ tzRF is the baseband equivalent output of the PA and )(~
tz is the baseband
equivalent input of the PA. ( )* is the complex conjugate.
The system is called a memoryless model if the coefficients b2k+1
are real, which
models the bandpass AM-AM characteristics and can be obtained by
single-tone AM-AM
power sweep measurement. The models in [50, 51] belong to this
genre. It is called a
quasi-memoryless model if the coefficients b2k+1 are complex, which
models the bandpass
AM-AM and AM-PM characteristics and can be developed by polynomial
fitting to
single-tone AM-AM and AM-PM power sweep measurement data. By
incorporating the
AM-PM conversion effects in the quasi-memoryless models, the effect
of phase shift in the
output waveform with different input power levels is accounted for
appropriately. The
models in [6, 7, 52, 53] are the examples of quasi-memoryless PA
models. The memoryless
model is simple to develop and the distortion can be directly
related to the model
parameters. However, the use of memoryless models has restricted
conditions. First, the
signal bandwidth should be narrow compared to the input and output
filters so that they can
be viewed as flat filters to the bandpass signals. Second, no
odd-order distortion should be
30
produced from even-order nonlinearities in the active devices [49].
The memoryless
models cannot take into account the linear and nonlinear memory
effects and the use is
limited in wideband and multi-channel systems.
It is well known that PAs present linear memory effects at the
input and output of the
devices due to the input and output tuned networks. Besides these
linear memory effects,
there are some dynamic memory effects resulting from the second
order interactions, the
active-device low-frequency dispersion, electrothermal
interactions, and bias circuitry
[56–58]. These dynamic memory effects show up only in nonlinear
regimes and are called
long-term memory effects. Extensions from the memoryless
nonlinearities are needed to
model the memory effects due to the bandpass system bandwidth
limitations. A filter
before and after the memoryless block that results in two-box or
three-box models are
necessary to account for the shifts in AM-AM and AM-PM plots [14,
54]. More
complicated model structures are developed to handle the nonlinear
memory effects arising
from the second order interactions, active-device low-frequency
dispersion, electrothermal
interactions, and bias circuitry [9–13].
As the advance of the PA behavioral modeling techniques, it seems
that the quality of
the models is more dominated by the parameter extraction process
than the model structure
itself. With the models being more sophisticated, the parameter
extraction process is
getting more complicated and often direct parameter extraction is
unable to be achieved
[49].
2.3.2 Behavioral Modeling for I/Q Quadrature Modulators
Direct conversion I/Q quadrature modulator plays an important role
in RF systems by
31
performing quadrature modulation and up-converting. It introduces
nonlinear distortion
mainly from the baseband nonlinearities and linear distortion as
well due to carrier leakage,
gain and phase imbalances. The impairments in the quadrature
modulators can degrade the
performance of wireless communication systems. In recent
literatures, modeling
techniques to capture circuit-level behaviors of RF mixers have
been introduced. The
authors in [4] develop scattering parameters based models for
mixers for use with vector
network analyzers to characterize the linear frequency translation
behaviors. Reference [1]
presents a frequency domain memoryless nonlinear behavioral model
for RF mixers. A
linearized mixer model as a conversion matrix [2, 3] was developed
to characterize both
the frequency conversion signals and the main mixing products and
the LO leakage. It has
limitation due to the intrinsic input to output linearity of the
conversion matrix which leads
to lack of accuracy in predicting the spectral regrowth. These
works focus on the
characterization of detail performances of mixer blocks. A
behavioral model to describe
the system level performance of RF quadrature modulators is however
necessary to
facilitate RF systems design. In [38], the authors modeled the
frequency-dependent
gain/phase imbalance and the DC offset by constructing channel
models and using
lease-square estimation to obtain the model parameters in order to
pre-compensate the
impairments in the digital predistortion linearization system.
Huang et al. [39] estimated
the gain/phase and DC offset by applying least-square-based
technique to the measured
instantaneous power at the input and output of the transmitter. In
[47], the authors modeled
a quadrature modulator nonlinear responses by extracting the model
parameters with a
pulsed DC signal input. The authors in [59] presents a time-domain
baseband equivalent
32
behavioral model for RF mixers for system-level characterization,
however, it only
captures the AM-AM nonlinear responses of the mixer and it is
desirable to have a
frequency domain behavioral model for ease of simulation. To the
best knowledge of the
author, there is no research work on system-level behavioral
modeling of the I/Q
quadrature modulators that completely characterizes both linear and
nonlinear distortion,
which motivates the present work.
2.4 Bandpass Nonlinearity Analysis
The signals in RF systems are narrow-band and band-selected signals
and therefore
the bandpass nonlinearity concept is widely used in the analysis of
nonlinear RF systems.
The concept of bandpass nonlinearity was developed in 1950s by
information theorists in
order to simplify the analysis of the impact of the nonlinear
circuits on the degradation in
C/N when a modulated RF carrier passes through the nonlinear
circuits followed by a
bandpass filter centered at the carrier frequency [60]. A complex
envelope representation
of a modulated carrier signal can be expressed as:
[ ] tjtj c
2 1)(cos)()( * . (2.25)
where A(t) and )(tθ are the amplitude and phase modulation
components, cω is the RF
carrier, and ( )* is the complex conjugate.
The carrier modulation is contained in the complex envelope, )(~ tz
, which can be
represented in either polar or rectangular form
)()()()(~ )( tqjtietAtz tj ⋅+== θ . (2.26)
33
where i(t) and q(t) are the in-phase and quadrature components of
the baseband signal. The
bandpass nonlinearity can be modeled and analyzed using complex
power series [61],
complex power series combined with statistical techniques [6, 62],
Chebyshev
transformations [63], Volterra series analysis [64], and etc.
2.4.1 Bandpass Nonlinearity for Power Amplifiers
The bandpass nonlinearity concept simplified the analysis for power
amplifiers by
eliminating the need to consider other nonlinear terms harmonically
related to the carrier
frequency since all the distortion components except the ones
centered at the carrier
frequency are eliminated by the bandpass filter [5, 15], as shown
in Figure 2-12 where w(t)
is described by Equation (2.25). Note that the application of
bandpass nonlinearity requires
that the modulation bandwidth is narrow compared to the carrier
frequency so that the
observed output spectrum is not affected by distortion terms from
other harmonics related
to the carrier.
w(t) G~ )]([~ twG
tj c c
etzG ω ω )]([~
cωcω
Figure 2-12 Block diagram of the bandpass nonlinearity of a power
amplifier.
A simple way to characterize the bandpass nonlinearity for power
amplifiers is to use
the quasi-memoryless model, i.e.
+ka characterizes the instantaneous amplitude and phase responses
of the
34
nonlinearity. By a couple lines of algebra on binomial expansion,
the complex envelope of
the first zonal filter output [ ])(~~ tzG cω (the output terms
center around the carrier frequency
without all the other higher order terms) is [5]:
[ ] [ ] [ ]∑ =
++
+
+ =
+ka is to do an AM-AM and
AM-PM measurement or simulation followed by a least-square curve
fitting. Note that the
fitted coefficients 12 ~
+
+ = +
2.4.2 Bandpass Nonlinearity for Quadrature Modulators
The bandpass nonlinearity of quadrature modulators differs from
that of power
amplifiers as a result of the fact that the dominant nonlinearities
in integrated quadrature
modulators are the baseband transconductors. Assuming the I and Q
channels are isolated,
a bandpass nonlinearity representation for a direct-conversion
quadrature modulator is
shown in Figure 2-13. Because I and Q baseband transconductors
process the baseband
information, the outputs of the I and Q channel nonlinearities, [ ]
)(~ tiGI and [ ] )(~ tqGQ
contain the dominant instantaneous amplitude and phase response,
which are subsequently
up-converted to the RF spectrum terms around the carrier frequency.
Different from the
power amplifiers, the I and Q channel outputs after the first zonal
filter, [ ] )(~~ , tzG II cω and
[ ] )(~~ , tzG QQ cω include the upconverted instantaneous
responses out of the I and Q baseband
35
nonlinearities, and so does the final quadrature modulator output,
[ ] )(~~ tzG RFcω .
wcw
ω )(~~
IG~
Figure 2-13 Block diagram of the bandpass nonlinearity of a
quadrature modulator.
2.5 Multisine Signal Modeling
Because the development of behavioral models usually relies on
input-output
observations, the selection of signal excitations used for
parameter extraction and
validation plays an important role in determining the quality of a
behavioral model.
Traditionally measured or simulated AM-AM and AM-PM characteristics
based on single
tone excitation is used to model narrow band nonlinear circuits. A
single tone excitation is
simple and eases the development of the behavioral models. However,
it has its limitations.
First, only odd-order nonlinear characteristics can be modeled by a
single tone AM-AM
and AM-PM data. The even-order nonlinearity can lead to important
memory effect which
manifest as IM3 asymmetry. Due to the incapability of a single-tone
excitation, two-tone
excitations were used to capture the even-order nonlinear [9, 57].
Second, a single-tone
measurement, or even a two-tone excitation, is insufficient to
model a wide-band nonlinear
circuit like a multi-channel power amplifier because the
characteristics of a wide-band
36
nonlinear system are different at different frequency, which
demands more complicated
signal excitations in order to capture the wideband
characteristics. Recently much attention
has been paid to multisine as the excitation, especially with the
aid of large signal network
analyzer (LSNA), in order to model the power amplifiers [12,
65–68].
A multisine signal excitation consists of N equally spaced
sinewaves, with random or
constant amplitudes and phases. Compared to digitally modulated
signals, they are easier
to generate and simulate by most time/frequency domain simulators.
These features render
multisine stimuli considerable usefulness not only in behavioral
modeling, but also in the
areas of characterization of spectrum regrowth both out-of-band and
in-band [15, 69–71],
device or system testing, and calibration of nonlinear-vector
network analyzers [72].
Due to the increasing importance of multisine excitations, in this
section an approach
for generating multisine to accurately represent digitally
modulated signals was presented.
Tradeoffs in the number of tones required to accurately represent a
particular number of
symbols for a given system are discussed. A multisine
representation of a CDMA reverse
link signal was designed and verified through measurements of ACPR
and EVM when the
signal is applied to a nonlinear amplifier.
2.5.1 Generation of Bandpass Multisines
Remley [69] examined four types of multisines each with different
magnitude/phase
relationships to approximate digitally modulated signals in ACPR
measurement. Among
the four multisines, although the multisine with a closely matched
peak-to-average ratio
(PAR) to the digital signal had better ACPR results, none of the
multisine signals
accurately predicted the ACPR of the actual digital modulation. In
[73], the authors
37
designed their bandpass multisines by matching the output power
spectral density or higher
order statistics of a digital signal in a general nonlinear dynamic
system. One limitation of
this approach is the expensive computation cost of matching the
higher order signal
statistics even with small amount of tones. The multisine signals
from both efforts were
used to primarily model out-of-band spectral regrowth distortion.
They are not easily
applied towards predicting waveform quality figures-of-merit such
as SNR and EVM. In
this section, an easy and straightforward approach based on
discrete Fourier transform
(DFT) transformation was presented to generate a multisine which
can accurately represent
a digital communication signal.
Construction of multisines from Fourier coefficients begins by
defining a bandpass
multisine as a finite sum of sinusoids with unique amplitudes and
phases
( 1) / 2
( 1) / 2
x t A f j f tπ θ −
=− −
= ⋅ + ⋅ ⋅ + ∑ . (2.30)
where N is the number of sinusoids, Aj is the amplitude, fc is the
carrier frequency, f is the
frequency spacing, and jθ is the phase. The design goal is to
determine the four parameters,
N, Aj, j , and jθ .
∑ −
=
nx π n=0,1,2,…,N-1. (2.31)
where X(k)/N are the normalized DFT coefficients of x(n). Equation
(2.31) can be
rewritten in the form as
38
+
=
= ⋅∑ n=0,1,2,…,N-1. (2.32)
where X(k)/N =Ak.ej kθ . This means a discrete signal x(n) is
equivalent to a sum of
sinewaves whose amplitude and phase are defined by the DFT
coefficients of the signal
and frequency spacing is determined by the frequency resolution of
the DFT. Thus a
multisine representing sampled signal can be constructed in a
straightforward way by using
the DFT coefficients to define the amplitude, phase, and frequency
spacing of the tones.
One drawback to use DFT coefficients is that a signal with N sample
points will yield
N DFT coefficients that results in a large number of tones required
to represent the sampled
signal. However, careful examination of the power spectrum of the
RF signals shows that
the majority of the total signal power is contained within a small
fraction of the total
bandwidth of the DFT, as shown in Figure 2-14 which is the power
spectrum of a typical
IS-95 reverse-link signal. The information content out of this band
energy is negligible.
The total number of tones necessary to represent the signal is
greatly reduced if the
out-of-band coefficients can be neglected, which give us a possible
solution to design a
multisine by truncating the bandwidth of the original signal.
39
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
-5 -4 -3 -2 -1 0 1 2 3 4 5 Frequency Offset from Carrier
(MHz)
Po w
er S
pe ct
um (d
B m
Figure 2-14 Power spectrums of IS-95 reverse-link signal.
The impact of truncating the bandwidth was quantified by
investigating the RMS error
introduced by discarding the out-of-band spectral components for
different normalized
truncation bandwidth. The normalized truncation bandwidth is
defined as the ratio of the
truncated bandwidth to the modulation bandwidth of the information
signal. For instance,
the signal modulation bandwidth of a reverse link IS-95 signal is
1.2288 MHz. The
truncated bandwidth was decided by comparing the root-mean-squared
(RMS) percentage
errors between the real IS-95 signal waveform and the equivalent
multisine waveform for
different truncated bandwidths.
The RMS percentage error was calculated as:
40
rms 100(%)ERR . (2.34)
where wk is the original signal waveform, xk are the time samples
of the multisine signal
waveform obtained from an inverse DFT (IDFT), and N is the total
number of time points.
For the IS-95 reverse link signal, the RMS percentage errors for
different normalized
truncated bandwidth are shown in Figure 2-15. The RMS percentage
error decreases with
an increase of truncated bandwidth and it is acceptable around the
normalized truncated
bandwidth of 1.2, which results in a 240-tone multisine
representing a 163 sµ IS-95
reverse-link signal.
Figure 2-15 RMS percentage error vs. normalized truncation
bandwidth for IS-95 signal.
The spectrum plots of the IS-95 reverse link signal and its
multisine equivalent
constructed with 240 tones are presented in Figure 2-16, which
clearly show that the power
components out-of-truncation BW have been discarded in the
multisine representation of
the IS-95 reverse link signal.
41
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
-5 -4 -3 -2 -1 0 1 2 3 4 5 Frequency Offset from Carrier
(MHz)
Po w
er S
pe ct
um (d
B m
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
-5 -4 -3 -2 -1 0 1 2 3 4 5 Frequency Offset from Carrier
(MHz)
Po w
er S
pe ct
um (d
B m
(b)
Figure 2-16 Power spectrums: (a) IS-95 reverse-link signal; (b) the
equivalent multisine.
Since the frequency, amplitude, and phase are available from the
DFT coefficients and
sampling frequency, the key design parameter is the number of
sinewaves needed to
accurately represent the original signal. The number of tones is
decided by two factors: the
resolution BW that is inversely proportional to the length of the
signal, and the truncated
(a)
42
bandwidth. For example, if 1 MHz bandwidth is truncated for
reconstruction of signals and
the frequency resolution is 10 kHz, the number of tones will be (1
MHz)/(10 kHz)=100.
In generating the multisine for the IS-95 reverse-link signal, the
signal length was
decided by monitoring its statistical properties. The length needs
to be long enough so that
statistical properties of this realization such as probability
density function (PDF), power
spectrum density (PSD) should have a good match with a
long-duration signal. In this work
a signal length of 163us (i.e. frequency resolution is 6.144 kHz)
was used to reproduce a
multisine for the IS-95 reverse-link signal, corresponding to 800
sampling points with a
4x1.2288 MHz sampling frequency. The signal length was selected to
make the frequency
resolution be an integer to avoid frequency errors. The frequencies
of the bandpass
multisine were calculated by a frequency translation from the
baseband to carrier
frequency [74].
2.5.2 Measurement Results for IS-95 Multisine
The accuracy of using the multisine signal representation was
assessed by
measurements comparing an IS-95 reverse-link signal to its
equivalent 240 tone multisine
representation, with a normalized truncation bandwidth of 1.2, when
applied to a 2GHz
nonlinear power amplifier. ACPR and EVM were measured and compared
for both
excitations over a range of output power levels. The IS-95 standard
defines the ACPR as
the ratio (in decibel) of the distortion power in 30 kHz, offset by
885 kHz from the carrier
frequency, and the power in the desired channel with a bandwidth of
1.2288 MHz [75].
Measurement was conducted using a vector signal analyzer (VSA) to
measure the power
spectrum and EVM of the two signals. It is worth mentioning that to
maintain consistent
43
measurements between each signal it was necessary to trigger the
measurement equipment
at the beginning of each signal for the same time duration of
approximately 200 data
symbols.
With IS-95 reverse-link signal excitation, the 1 dB compression
point of the power
amplifier occurs at about 13 dBm output power level, as shown in
Figure 2-17. The input
power sweep is from −22 dBm to 6 dBm, which corresponds to 0 to
17.3 dBm output
power range, fully covers the weak and strong nonlinear regions of
operation for the device.
Notice that the gain compression characteristics of the IS-95
signal are indistinguishable
from the multisine signal because the PAR of these two signals that
is determined by the
spectrum in the main channel are very similar.
Figure 2-17 Gain compression characteristic of the PA for IS-95 and
multisine excitations.
The IQ data of the IS-95 reverse-link signal were loaded into
Agilent E4438C signal
generator and carrier modulated to 2 GHz. The multisine signal was
loaded into the signal
generator using Agilent Enhanced Multitone signal studio that
defines a multisine signal
44
by the power, phase, and frequency of each tone.
ACPR measurement results are presented in Figure 2-18. The ACPR
measurements of
IS-95 signal and its equivalent multisine match very well when the
power amplifier goes
into nonlinear region, especially when the PA approaches
saturation. In the linear and weak
nonlinear region, there are discrepancies between the two
excitations at low output power
levels, which are due to the truncation of the out-of-band
components in the multisine
signal (i.e. the out-of-band rejection of the multisine is greater
than the original IS-95
signal).
Figure 2-18 Comparison of ACPR for IS-95 and multisine
excitations.
The EVM measurements of the two excitations are shown in Figure
2-19. There is
excellent agreement between the measured EVM of the IS-95 signal
and multisine
representation applied to the nonlinear amplifier, which
demonstrates that the truncated
45
multisine signal is an equivalent representation of the original
IS-95 signal. Since EVM is
related to the in-band distortion, therefore, the multisine can
accurately predict the in-band
distortion of the original signal. The EVM decreases from the first
maximum point because
the PA enters the saturation region. The minimum EVM is limited by
the measurement
noise floor of the signal source and the dynamic range of the VSA
equipment.
Figure 2-19 Comparison of EVM for IS-95 and multisine
excitations.
Compared to other approaches such as signal statistical property
matching, this DFT
based multisine generation method presents an efficient and
straightforward way for
generating accurate multisine signals for characterization of
nonlinear circuits. An IS-95
reverse-link signal was employed to demonstrate this approach.
Measured ACPR and
EVM of an IS-95 and equivalent multisine signal applied to a
nonlinear power amplifier
are in excellent agreement. The results showed that the DFT
generated multisine can very
46
accurately predict the ACPR and EVM of the original digital
modulated signal.
2.6 Summary
In this Chapter the properties of a quadrature modulator and its
three different
applications in modern wireless communication systems were
reviewed. Three major
physical impairments of quadrature modulators, the DC offset, the
gain/phase imbalances,
and the nonlinear distortion were introduced and their general
impacts on system
performances were discussed, which motivates the research to
develop system-level
techniques to accurately and efficiently characterize these
quadrature modulator
imperfections for system design and verification. Behavioral
modeling techniques for this
use were reviewed and the underlying bandpass nonlinearity concept
developed mainly for
power amplifier modeling was extended to quadrature modulator
modeling. The
differences of the bandpass nonlinearity representation for power
amplifiers and
quadrature modulators were analyzed. In the last section a
particular type of signal,
multisine signal, was successfully modeled to accurately represent
the CDMA digitally
modulated signals.
Modulators
Looking in frequency domain, physical impairments of quadrature
modulators such as
nonlinear distortion, carrier leakage, gain/phase imbalances and
noises produce undesired
spectrum contents which degrade the signal quality. The undesired
signal contents consist
of two components. One is correlated to the desired signal, which
causes gain compression
or expansion. The other component is uncorrelated to the desired
signal that results in the
degradation of signal EVM [8]. As mentioned in Section 2.2.3, the
SNDR is defined as the
signal power to the noise and uncorrelated distortion power for
characterization of the
system performance degradation due to the nonlinear distortion. In
the section, the SNDR
definition was expanded as the ratio of the desired signal power to
the noise and the
48
uncorrelated interference power including not only nonlinear
distortion but also image
spectrum and carrier leakage, in order to characterize the
degradation of the quadrature
modulator performances caused by these physical impairments. With
this definition, the
SNDR can be calculated by performing cross-correlation between the
desired and the
⋅
⊗ −
⊗
. (3.1)
where yact is the actual signal and yideal is the desired signal. ⊗
denotes cross-correlation.
The numerator 2
ideal act y
y y is the portion of power in the actual signal that is
correlated to the desired signal because
⊗
yy is the magnitude of the actual
signal projected to the direction of the ideal signal, the square
of which gives out the
correlated part of power of the actual signal.
The denominator
y is the portion of power in the actual
signal that is uncorrelated to the desired signal. Rewrite ideal
ideal
idealact y y
It can be seen from Equation (3.2) that ideal ideal
idealact y y
⊗ 2 is the production of the direction
of the ideal signal and the magnitude of the actual signal vector
projected to the direction of
the ideal signal, which is the vector component of the actual
signal projected to the
direction of the ideal signal. Therefore, ideal ideal
idealact act y
⊗ − 2 is the vector component of
the actual signal that is perpendicular to the direction of the
ideal signal, which is
uncorrelated to the desired signal.
ideal ideal
−= 2 . (3.3)
The square (auto-correlation) of (3.